Haven't these people heard about HIPAA? You can't just transmit personal health information over arbitrary text message networks. It doesn't matter if they have a "secure server" somewhere(*). The whole network needs to be secure and auditable. And something tells me that this isn't the case for text messages.
(*) Google can find exactly two mentions of text4baby and HIPAA, both of which just say that there is a secure server.
There is no time like the present for postponing what you ought to be doing.