First off, make sure you're running Dynamic C 10.72D as advised by Campbell in another answer. The original Dynamic C 10.72 release doesn't support TLS 1.2, which is required by almost all web servers now.
What exactly did you use in the httpc_set_tls() function? SSL_F_PEER_CERT_OK on its own, or (SSL_F_REQUIRE_CERT | SSL_F_PEER_CERT_OK)?
If you uncomment the define for SSL_CERT_VERBOSE, does the sample print out useful information on why it rejected the connection? Is it possible this is something other than a certificate failure?
You need to take a look at the certificate used on https://api.telegram.org (by using a web browser), and find the root certificate that signed it. Configure the HTTPS_CLIENT sample with that root certificate and you should be able to make connections.
Take a look at the Remote Program Update library and samples for methods of doing remote firmware updates for long-term support of your product. You could also use the HTTP client code to download new root certificates as necessary. You could configure a small FAT file system on the serial flash of the RCM67xx and store root certificates there.