Not push button. With a caveat. The scenario you described leaves quite a bit of fudge factor. You're a couple days of normal walking from the trailhead, and sick. If you wait, and you fell better in a day and a half, you can get going, and then you'll be maybe a day or a half day late getting to the trailhead. call home the soonest you can, and you'll hopefully stop the show before it starts. My answer might be different if you were 5 days of waking from a trailhead (which you might be able to pull off up there in the Frozen North)and still feeling sick with only a day left, then the situation would be different. Or the other way, if you're only 1 day from the trailhead, maybe you try to stagger out no matter how crappy you feel. Different scenarios, different answers.
All of it brings up an interesting thought - perhaps this is a case for the SPOT, using the tracking feature and the customized meassage. If you keep the tracking on while you are stopped, the folks at home can see you are not going anywhere. Then, if you had a pre-programmed message telling them you'll be late but okay, you send that and all is good. Assuming, of course, that your SPOT makes contact with the sattelite. How bold an assumption this is with the new version of the SPOT is yet to be seen.
It's also a reminder of why it is a good idea to be conservative in your planning of days and mileage, giving you time to deal with something like this if it happens. And it does happen. My son got sick this summer and we had to bail in the middle of a trip, after sitting out a couple days while he was sick. Still don't know what hit him. He could not have gone anywhere for a day or two - fainted at one point and was weak as a kitten.