Job loss is always a qualifying event, but there is a practical processing delay. If you were to sign up on December 28, there is absolutely 0 chance the paperwork would get done in time to have you covered by January 1.
If you were to sign up, say, on December 20, well maybe if the insurance company is responsive, but you're taking a risk. Long story short: sign up today.
This is why I always recommend people make their last day early in the month, if possible. Most employers cover departing employees through the end of the calendar month, which leaves time to sign up, get processed, clear up any hurdles, and not have any lapse in coverage.