Will I get a tax penalty if I do not take Cobra (regular insurance ended on 4/7/17 with prior employer, and new insurance through my husband's plan begins 6/1/17)?

No.

You are allowed a 'short gap' of up to 2 months, one time per year.  Insurance for at least one day in a month is considered an 'insured' month, so your only 'uninsured' month is May.  That qualifies for the 'short gap' exemption.

When you tell TurboTax which months you had insurance, it will automatically apply that exemption to the penalty.