This is a constitutional crisis.

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42. Nothing in the constitution precludes one side from seeking the complete capitulation of the other

Heck, when Bill Clinton vetoed both a debt ceiling increase and a CR and the government went into shutdown mode, he would have been perfectly happy to accept the complete capitulation of the houseand senate to his position. In the end, both sides compromised (the house and senate moreso than Clinton), but there was no consitituional issue as whether it was appropriate for Clinton to veto funding and debt ceiling bills that had been passed by a majority of both the house and senate.

Also, while the government shutdown is bad for the economy and bad for millions of people who don't have certain government services available to them, it is not a total shutdown of the government. The House and Senate have not ceased functioning, the President is still carrying out his constitutionally assigned roles as chief executive and commander in chief and the Supreme Court is still open for business.

Most of what makes up the government is unquestionably essential, but its also mostly not constitutionally mandated. If the House and Senate wanted to abolish the Department of Homeland Security and the President supported that action, it would be a constitutional action.

So what we have here is a crisis, but not necessarily one that I would describe as a "constitutional" crisis.