Sorry to hear about the job loss. These are difficult times for a lot of people.
My understanding of verification of employment is that it relates to a specific application request for credit. On a credit card, I doubt a bank would want to incur fees to obtain that information.
For your long term credit health, I would advise only using the card for items you know you can pay immediately. I wouldn't bother with updating your income to CapOne, because no credit apps should be considered for now.
Good luck in your job search.
@Dirge wrote:Do they know I lost my job? I know there are databases that creditors can use to check employment.
Or maybe it's just a coincidence, though it doesn't seem like they should need an update since the one I gave them in April.
It is not really a coincidence. It is just your paranoia kicking in. It happens. 
You could receive this email any other time you have a job, but you would think differently. It is just as they said: "..so we can help you find eligible offers." I would not think much of it.
In any case, I would just not respond. It's very unlikely that they know you just lost your job. I've had a job loss and salary cuts before. I just never update my salary until it's higher.
I don't have a Capital One card, but this appears alot on my LL Bean Card which is Citi and I have never updated it. I think you are probably ok to ignore it. I think I have seen it once in a while on my Amex accounts as well. I wish you well in your job hunt.
