Putting a quota limit or asking people to justify why they should get a free shoot is a PR nightmare. People are going to get turned down and regardless of the way the decision is made they won't be happy.
I would suggest that instead of unlimited free, you offer a discount to military, which enables you support them while at least covering costs. Then, offer unofficial free shoots to needy people like the case in the OP. This gives help where it is needed without creating a feeling on entitlement in others. Then, if you want, volunteer as mentioned in the posts above, where you are one of several volunteers and the load is shared.
I agree with a lot of this. I certainly wouldn't want to be the one to decide who "deserves" a free shoot because of their hardship, and who isn't "needy" enough. Pairing with an existing charity or organization would take a lot of the difficulty off your shoulders, and allow you to focus on the photography.
My wife lost her job several months ago, and then was injured in a car accident and can't work for more than a few hours at a time. Our income is about half of what it was a year ago, and we've had to cut back on almost all discretionary spending. Does that make us "needy?" I suppose if things get really tough, I can sell the BMW.