Well that error is common in Wordpress and it is a hosting problem. It usually meas that the  webserver can't figure out the invalid response from a requested page.

That could mean a lot of things, a slow server or db is a possible reason. Could also be a DNS problem. Could be something been offline (a db?).

I highlt doubt it has to do directly with wpForo.

For the last 3 weeks I have been working with my hosting company trying to figure the cause of the 502 Bad Gateway error. The host company says it is not a hosting issue.

The 502 Bad Gateway error is only seen by old logged in users when Forum Cookies is Enabled. If users mark "all read" it sometimes fixes the problem for some users.  Any expertise would be appreciated. 

This is a browser-specific issue, so the rate of such a problem is too small. It can be solved if you set wpforo_max_logged_topics value to 50 via according filter hook. Please put this code in your active theme functions.php file:

function my_wpforo_max_logged_topics( $num ){
    return 50;
}
add_filter('wpforo_max_logged_topics', 'my_wpforo_max_logged_topics', 12);

function my_wpforo_cookie_max_logged_topics( $num ){
    return 50;
}
add_filter('wpforo_cookie_max_logged_topics', 'my_wpforo_cookie_max_logged_topics', 12);

How to Easily Add Custom Code in WordPress (without Breaking Your Site)

Or you can disable all forum cookies in Dashboard > Forums > Tools > Privacy & Rules admin page. But the first solution is better.