As your problem seems to be relative to the Linux OS, I'm afraid I can't really help you neither. But, just in case...
1/ Which version of Indy are you using ?
Is it a recent one ? The current version is 10.6.2 rev 5301: it can be downloaded from
https://indy.fulgan.com/ (in the ZIP directory).
2/ Which version of Open SSL are you using on your Linux computer ?
Is it possible that the version you've used for your tests on Windows is more recent than the one on Linux ? Do you use an old one in Linux (i.e. with known security issues) ?
AFAIR, Indy "deactivates" some old versions known to be unsecured. The current versions are 1.0.0s, 1.0.1p and 1.0.2d (see
http://openssl.org/), though the recent ones are still accepted by Indy anyway.
BTW, have you tested that Open SSL is found and correctly loaded in Indy ?
uses ..., IdSSLOpenSSL, IdSSLOpenSSLHeaders;
...
ShowMessage('Open SSL version found = ' + OpenSSLVersion + sLineBreak +
'Failed = ' + WhichFailedToLoad());
...
3/ Finally, you can still try to post in the Indy forum and describe your problem. I guess that Remy Lebeau (the current maintainer of the Indy library) will probably be able to help you, or at least to provide you some clues.
Indy forum :
http://forums2.atozed.com/