as a J1-student you are exempt from counting days for the substantial presence test for the first 5 calendar years, so 2016 would be your 3rd year as a non-resident alien. Assuming you are staying until 2021, 2017 would be your fourth and 2018 the last. In 2019 you would start using the substantial presence test to determine residency. The 5 years are a lifetime exemption, so once they are used up, you cannot use them again (it's different, if you switch to a J1 nonstudent visa, as a postdoc for example).
As a nonresident alien you need to file form 1040NR plus form 8843 (plus a state return). You cannot file a joint return as nonresident aliens. You would both file as married filing separately.
Turbotax does not support these forms, so you cannot use Turbotax. You can download all forms from the IRS website. Some universities provide access to tax preparation software for their foreign students.
Have a look at publication 519: https://www.irs.gov/publications/p519/
For tax purposes, you are not required to have health insurance. But it might be advisable to get health insurance for your spouse, health care is very expensive in the US.
