The short answer is, yes, you can claim a child who was born alive as a dependent, even if they died shortly after. A Social Security number is not absolutely required, but the tax return will go through easier if you have one.
One important factor mentioned above is that you must file an original return listing the child and it must be “on time.“ On time means that you file before April 15, or that you get the extension and file before October 15. If you file a return that does not list the child and then you file an amended return later to add them, or if your tax return is considered late because you did not get the extension, then the child tax credit may not be paid.
Separately from the dependent issue, you and your spouse can list out-of-pocket medical expenses on schedule A as an itemized deduction. You may or may not get an actual tax benefit depending on your other deductions and tax situations.
