Have to say I'm starting to wonder if the Republican strategy could actually null out its self-destruction. The longer this goes on, the more the blame spreads to the Democrats, at least from the CBS polling I've seen on the news since the shutdown began. The number of respondents blaming Democrats is up by ~10%, although there's a still a significant majority blaming Republicans. (the number blaming "both" has also risen from single to double digits).
Right now the Democrats have the moral high ground, but all it takes is one dumb slip-up in their management of the situation and they could easily look just as bad. (for instance, if they sunk to the GOP's level and made similar demands from the left).