My husband is a UK citizen and I am American.  We currently live in
the US with our 3 year old twins, but we will be moving for 9 months
to France for a contract position that I have received.  Through my
job, I will be insured, but I need to provide proof of coverage for my
family.  Will my husband's UK citizenship still enable him to
coverage?  Our children coverage?  If so, what forms would he need? 
Regarding Visas, since he is European, do we need to apply for visas
through the US, or do we have the option of processing through the UK?
Which is easier and/or less expensive?  Thank you for your help!

As a UK citizen, your husband qualifies for French health care
- IMO rather better than the UK
- there is some silly little form one needs to get from a Post Office in the UK

Not sure about the children, if they are 'UK residents' then it should be Ok
- the same reciprocal rules

I'm a bit curious that you 'need to provide proof of coverage'
- under those circumstances I would look for an inexpensive insurance
policy where the vendors know the French rules.

Possibly someone else is more current with the hoops.

Your husband is probably not covered, and you should obtain further
advice and if necessary insurance.

Reciprocal cover in the EU is given for those who are normally
resident in another EU country - residence, not citizenship, is the
issue because the scheme depends on you paying the relvant social
security payments in your normal country of residence.

The card you need to obtain is an EHIC card (www.ehic.org.uk). 

If during your stay in France you are being paid in France (and not
the US) and paying French social security contributions you will be
covered by the French scheme anyway,

Be aware that the French medical scheme does not cover all costs.