Forum Moderators: open
Ask the guy politely to back off a little on the self-promotion. Point him to your Terms of Service (which you may need to write first if you don't have one). Always best to have a policy on this, and implement it at as early a stage in a forums development as you can.
If you allow "signatures" in posts, and people are abusing them, disable them and allow a member "profile" instead. Signatures tend to encourage spammers on popular forums. That's one of the reasons sigs are not allowed on WebmasterWorld. Pointless repetition is the other.
You risk allowing links to "bad neighbourhoods", so use a CGI redirect script on the profile page for the users link. Or put profiles in a '/profiles/' directory (or mod_rewrite to effect the same) and use robots.txt to stop them being spidered.
Assuming all is spiderable, and you don't use any kind of redirect script, or javascript, to "hide" the URL's, then yes they count as backlinks.
TJ
And note on this forum all the links to westhost.com on the pages.
google is not served the page you see, this has been discussed before, easy to verify by simply using the google navigator user agent string and browsing WebmasterWorld. The link to westhost is only for non spider consumption, there are no outbound links of any type on the main content pages of this site except to other phd software sites for spiders.
Trillianjedi is right I think, allowing signatures is pointless and potentially very damaging to your forum and site, if members want their site listed it can be listed in the members information, then you can disallow the members information pages to spiders to avoid possible bad neighborhood type penalties. Speaking just for myself, when I see avatars, signatures, etc, I'm annoyed at the clutter and waste of resources and page space. That seems like a holdover from the teenage years, at best.