I just noticed that you said "this year before the last day to file taxes." I thought you meant before January 1, in other words, was reading it as "If my babies are born right at the end of the calendar year, do they still count as deductions in that year?" The answer to that second question would be yes, a baby born between January 1 and December 31 of a given year, even if born in the last moment of December 31, counts as a deduction for that year.
But it doesn't have anything to do with the tax filing deadline for a given year, which is of course April 15 of the following year (in the U.S.) Filing deadline for taxes is three and a half months after the end of any activities regarding which you are filing, to allow you to collect all the data, the 1099's and stuff, and to notice if you have had any babies in the previous year. So even having a baby April 14 wouldn't affect anything, that baby will relate to the calendar year in which it is born, for purposes of tax deductibility.
Your baby has to be born during the year for which your filing. So this year were all filing our 2014 returns so the baby would have had to be born during that year.
I believe they have to be born before the first of the new year
Yes. When my son was born January 4, the only person chagrined that he had not been born five days earlier was my accountant.
You can claim them if they're born before jan. 1st