provided you say you've started the process by March 31st.

See: http://www.washingtonpost.com/national/health-science/obama-administration-will-allow-more-time-to-enroll-in-health-care-on-federal-marketplace/2014/03/25/d0458338-b449-11e3-8cb6-284052554d74_story.html

As long as you apply for an extension by about mid-April-ish, you will have an apparently indefinite period of time to sign up.

I understand they are trying to give people every opportunity to enroll, but I'm concerned that the precedent they are setting might make it easier for a future Republican administration to change the rules of the ACA by executive order in an attempt to undermine it. For example, what will stop them from simply suspending the individual mandate or tax penalty?

you need to check with your individual state...

I just want to make that clear, because people seem to be glomming onto the current deadline as being some sort of irrevocable finality. It's not. For people who didn't sign up this time around (and who don't qualify for the exigencies that would allow you to sign up between enrollment periods), they can do it in November. And for the first year, the penalty for failing to sign up by this current deadline is not draconian (in 2014: $95 per adult and $47.50 per child (up to $285 for a family) or 1% of your taxable income, whichever is greater).

who do not file tax returns, no penalty, correct?

You may qualify for an exemption if:

You’re uninsured for less than 3 months of the year
The lowest-priced coverage available to you would cost more than 8% of your household income
You don’t have to file a tax return because your income is too low (Learn about the filing limit.)
You’re a member of a federally recognized tribe or eligible for services through an Indian Health Services provider
You’re a member of a recognized health care sharing ministry
You’re a member of a recognized religious sect with religious objections to insurance, including Social Security and Medicare
You’re incarcerated, and not awaiting the disposition of charges against you
You’re not lawfully present in the U.S.

https://www.healthcare.gov/exemptions/

I have a friend who gets paid by his brother to stay home and care for mother with Alzhiemer's. So, he's being paid under the table, basically. He has no health ins., is 51, and living here in CA. His wife is a f/t student, doesn't work and also has no health ins

It will pay the brother to care for the woman, if she qualifies.

She doesn't qualify for aid, they already checked. Thanks though!