I don\'t think SSN will change the STATUS.
INS and SSA is two different organizations, I don\'t think your status will change if you take SSN. There are thousands of H4 having SSNs. SSA use to issue SSN wheather or not you are permitted to work in US, SSN was required to obtain Driver Licence, now the rules has changed in most of States and Issueing DL without SSN and stopped giving SSNs for H4s also.
If SSN changes the status what will happen to all the people who have SSN on H4 ?