I am on H1B visa.
During Jan - Sept last year my
wife (on H4 visa) was with me in USA and she had health insurance. In Sept I
changed job and at the same time my wife went outside USA to visit her parents.
I was not sure how long she would be outside US so I did not applied for her
health insurance from new employer. Recently in 2016 she has returned to USA.
She was not outside US for more than 330 days. So, in this case am I eligible
for any health insurance exemption during the time when she was not in US or I
will have to pay penalty?
Nothing in your post indicates you qualify for a waiver for the penalty, so you will need to pay the penalty unless you can find an exemption or hardship: