Was he in the military? Without a bit more on the names and dates, there is only so much volunteers can find and be sure that we are not leading you astray.
The only possible that I can find on SSDI (social security death index) is for a Jack Connery b. 30 Apr 1907 and died Oct 1986 in Legrand, Washington Co, Mississippi. It says he got his social security card in Illinois. Illinois is where the trail leads for the possible parents of John Emmert Connery. as there is a Emmett John Connery b. 1872 in Monroe, Greene Co. Wi. There are several family trees on ancestry for him and his family, except his youngest son, John E. Connery which have him born about 1907/8 and no death date.
If he was in military, you can search records. I've not had to do that myself, so I'm not sure of the steps that need to be taken, but, it can be done as well as checking on Social Sec. application on Jack in Mississippi.
This will be the tedious part of your search, hope you can find what you need, Best of Luck.