StartSSL will use whatever certificate digest algorithm you used in your certificate signing request. Most openssl.cnf files distributed with Linux OSes set the default algorithm to SHA-1 - that's nothing to do with Startcom.
Simply specify an explicit algorithm if you want to get a certificate using that. For example, if you do:
and give them that CSR, you will get back a SHA-256 certificate.
EDIT: They also have a SHA-256 root (in most browsers, though you don't need a second-preimage-resistant digest algorithm for a /root certificate/) and SHA-256 intermediates at https://startssl.com/certs/ - go to the relevant class directory and there is a sha2 directory inside that.
Simply specify an explicit algorithm if you want to get a certificate using that. For example, if you do:
$ openssl req -new -sha256 -newkey rsa:4096 -keyout foo.key -nodes
and give them that CSR, you will get back a SHA-256 certificate.
EDIT: They also have a SHA-256 root (in most browsers, though you don't need a second-preimage-resistant digest algorithm for a /root certificate/) and SHA-256 intermediates at https://startssl.com/certs/ - go to the relevant class directory and there is a sha2 directory inside that.