I think we can break down the legislative situation for the GOP into a few parts:
#1. Symbolic stuff. The Party of No strategy means trying to create as many tough votes for the Democrats as possible, while simultaneously giving conservatives Members of Congress as many good votes as possible to trumpet to skeptical activists back home. Good: Balanced Budget Constitutional Amendment (and, really, any constitutional amendments they can think of). Good: ostentatiously reading each bill. Good: fulfilling their (entirely symbolic and meaningless) pledge to insert a clause in bills saying why they're constitutional. Also good: investigations into malfeasance, real or imagined, in the Obama Administration and among liberals generally.
#2. Substantive stuff they really want to enact. I'm not talking about repealing health care, which they know they can't do and which they may or may not care about anyway; I'm talking about mostly bite-sized, doable,
#3. Stuff they have to do whether they like it or not. Chief among these are annual appropriations bills, and increases in the national debt ceiling.
#4. Unexpected stuff. Just worth mentioning that we can't really see everything that they'll have to handle.
Now, where are the weaknesses for Boehner?
First, many Tea Party and other GOP activists want confrontation for its own sake. That may make #2 above difficult, and may create huge problems for #3. Some Members of the Republican conference are going to find it very difficult to pass must-pass items, either because they honestly believe that lots of the normal things the federal government has done for decades are unconstitutional, or because they're terrified of primaries from half-crazed constituents who believe those things are unconstitutional.
Even worse, I do think that a fair number of activists want to refight the 1995-1996 shutdowns. As Boehner no doubt knows, everything about the politics of such a confrontation works badly for the House Republicans. Nevertheless, the leadership is vulnerable to charges of sellout on this, and they'll have to be disciplined enough to realize that actual shutdown doesn't end that problem.
Second, many Tea Party and other GOP activists passionately support contradictory and unpopular budget goals: balanced budgets with much smaller revenues and unpopular spending cuts. I'm reasonably confident that a unified front of GOP elites could satisfy these activists with symbolic stuff such as bans on earmarks and long-term goals "reached" with creative math, but as I said earlier there are clearly strong incentives for talk show hosts, backbench Members of Congress, and presidential candidates to stake out positions to the right of the GOP leadership in Congress, and they may well call Boehner on it.
Third, health care. Just as there were no good options for Democrats in marginal districts when health care was being passed, there are no good options for Republicans in marginal districts now. A just-plain repeal vote, full stop, leaves them vulnerable to being attacked over the many very popular things in the bill -- some of which are already implemented and would actually be noticed if they were repealed. Partial repeal leaves them vulnerable to charges of betrayal from activists. It's also actually difficult to find unpopular but significant slices of ACA to repeal. As for defunding, I think the WaPo story gets this right: it doesn't really do what the GOP wants it to do.
I see two paths on health care that would at least plausibly work for the Republicans. One is to do a repeal-and-replace vote in the House that involves passing an entirely nonworkable bill designed to include all the goodies in ACA, at least on the surface. It doesn't matter if there are no works inside; after all, the bill wouldn't stand a chance in the Senate, so it only needs to be designed to deal with attack ads, not the real world. The other, and better, path, is to just do a bill to eliminate the individual mandate. Of course, policy analysts will point out that without the individual mandate the ACA won't work -- but good government truth squadders would probably not give their seal of approval to attack ads saying that it's a vote against coverage for preexisting conditions.
Again, all of this analysis assumes that House Republicans choose to be the Party of No for the next two years. It seems to me that on that path the prospects for John Boehner depend on the extent to which GOP unity inside and outside the House can triumph over the impulse of everyone to prove themselves the "true" conservatives. Once that starts, Boehner has no good options; he's either going to wind up a RINO victim of a purge, or he's going to keep the approval of the fringe and make his conference massively unpopular with everyone outside the fringe.
But if they can stay as united in defense of phony symbolic actions by the Republican House as they were in opposition to phony threats of czars, death panels, gun control, and Black Panthers, then they'll all survive the next two years in pretty good shape.
*The necessary caveat: I'm still not at all convinced that Republican rejectionism in 2009-2010 has much to do with Republican success in the 2010 cycle, and may have done little more in most cases than to cost them in terms of policy outcomes.