Could a President Hillary have a Republican Congress for her entire Presidency?

[Never]

This scenario is definitely possible, but it depends on so many factors. For the most part, I'm going to put more focus on the Senate to answer this question, because that body seems more likely to switch control from one party to the other in the near future compared to the House (which could still flip by all means).

The Republicans could very well end up with at least 51 Senate seats this midterm, and Hillary Clinton's victory might end up being very close in 2016, but if Republicans just achieve a bare majority as a result of gaining six seats this year, there are plenty of Republican targets in 2016 will be up for reelection for the very first time. If Republicans lose just one of those seats (likely Sen. Kirk's in Illinois) without picking up any Democratic-held seats, it would give Democrats control of the Senate in a 50-50 tie, since the party would have the tie-breaking vice president's vote. Even if Republicans have 52 Senate seats going into 2016, there would still be the risk of losing the majority (Johnson could be the second Republican to fall that year). While Republicans could manage to pick up Nevada's seat if Sandoval decides to challenge Reid, that might not help the party keep control. On the other hand, a Republican Senate majority of 53 seats seems more able to withstand a Clinton presidency. Of course, if Republicans fail to take the Senate this year, it is nearly impossible for them to do so in 2016 during a Clinton victory. Regarding the House, a convincing win by Clinton could deliever it to her party, but this far out, I still suspect that Democrats winning the House in 2016 is not likely.

Now, if the Democrats do not win control of the Senate in 2016 along with a Clinton presidency, it is highly likely that Republicans will control both houses of Congress for the duration of the presidential term. For one, the president's party tends to do poorly in midterms, so the Democrats gaining control of the Senate would be difficult from the onset. The Democrats are quite overextended in the Class 1 Senate map for 2018, and there are five Democratic senators up for reelection in Romney 2012 states (WV, IN, MO, ND, MT). All of them could conceivably face tough fights to keep their seats (this might apply less to Manchin and Heitkamp than the other three), and it is not out of the question that Republicans could put races in states like Ohio, Virginia, Florida, and perhaps even Wisconsin into play. The only Republican seat that is at real risk of flipping to the Democrats is Dean Heller's in Nevada.  With this is mind, a narrow Republican majority in the Senate after 2016 could turn into a sizable one after it is all said and done in 2018. It would probably require a "rally around the flag" effect like a terrorist attack on U.S. soil or a very popular war abroad for Democrats to limit their losses in the midterm to a below-average number, and even if an event like this happened, there is still the chance that one or two Democratic-held Senate seats could fall into Republican hands.

In 2020, Senate control will depend on a few things. First, how many Senate seats will Republicans have to defend in unfriendly territory? If Ernst or Gardner win in Iowa and Colorado this midterm, they will probably have very difficult reelection races from the start. If Clinton proves to be a popular president, potential Republican senators in the South like Tillis and Perdue could have a tough fight, not to mention Mitch McConnell in Kentucky, who often has a difficult time in elections. On the other hand, a very good Republican midterm in 2018 could end up being a buffer in the Senate against a good Democratic year. Regarding House control, 2020 could be the year that Democrats win it if they haven't already, especially if Clinton is popular heading into her reelection campaign. The potential for this to happen could be muted by a lack of coattails affecting congressional races and Democrats being more prevalent in a smaller number of densely populated districts while Republicans hold the upper hand in less populated districts of greater quantity. Though the premise of Clinton being reelected to the presidency is certainly plausible, it is not a sure bet. It is also important to keep in mind that as of late, political parties have had trouble keeping the White House for three terms in a row, let alone four. 2020 is where the scenario of a Clinton presidency could end with the election of a Republican to the presidency.

Still, even if Clinton did win a second term in office, it could be all downhill from there relating to her party's strength in Congress. A president's second midterm usually hurts their party, and taking control of either house of Congress would be a herculean task if the Democrats haven't done so already. If they did hold the House and/or Senate after 2020, they could easily flip Republican depending on the circumstances surrounding the midterm. Republican's lost control of the Senate in the 2006 midterm after Bush 43's reelection, in 1986 after Reagan's landslide, and in 1958 after Eisenhower's reelection. It is definitely possible that Democrat's will lose control this year during Obama's second midterm. Similarly, I would expect Democrats to lose ground in 2022 if Clinton is halfway through her second term.

In short, there is a chance Hillary Clinton could experience four to eight years of Republican control of Congress, but the set of circumstances that would cause that are quite specific in my view. I could actually see Democrats winning control of the Senate in 2016, losing it in 2018, regaining the majority in 2020 provided Clinton is relected, and then losing it again in 2022, but it also isn't troublesome to envision the Democrats taking control of the Senate in 2016 and then losing it for the duration of Clinton's presidency after the first term, similar to her husband's loss of Congress in 1994. For Republicans to keep control of the House and Senate for her entire term in office, it would require a pretty good performance by the party this year and in 2018.