For my site now Boost module is working fine. It wasn't working though it was indeed generating the HTML files. I tracked it to .httaccess rule:
RewriteCond %{HTTPS} on [OR]
After commenting out it's working fine. But any idea why it is made to skip specifically HTTPS sites?
I will mention the D6 Boost .htaccess rules:
### BOOST START ###
# NORMAL - Cached css & js files
RewriteCond %{DOCUMENT_ROOT}/cache/perm/%{HTTP_HOST}%{REQUEST_URI}_\.css -s
RewriteRule .* cache/perm/%{HTTP_HOST}%{REQUEST_URI}_\.css [L,QSA,T=text/css]
RewriteCond %{DOCUMENT_ROOT}/cache/perm/%{HTTP_HOST}%{REQUEST_URI}_\.js -s
RewriteRule .* cache/perm/%{HTTP_HOST}%{REQUEST_URI}_\.js [L,QSA,T=text/javascript]
# Caching for anonymous users
# Skip boost IF not get request OR uri has wrong dir OR cookie is set OR request came from this server OR https request
RewriteCond %{REQUEST_METHOD} !^(GET|HEAD)$ [OR]
RewriteCond %{REQUEST_URI} (^/(admin|cache|misc|modules|sites|system|openid|themes|node/add|comment/reply))|(/(edit|user|user/(login|password|register))$) [OR]
RewriteCond %{HTTPS} on [OR]
RewriteCond %{HTTP_COOKIE} DRUPAL_UID
RewriteRule .* - [S=1]
# NORMAL
RewriteCond %{DOCUMENT_ROOT}/cache/normal/%{HTTP_HOST}%{REQUEST_URI}_%{QUERY_STRING}\.html -s
RewriteRule .* cache/normal/%{HTTP_HOST}%{REQUEST_URI}_%{QUERY_STRING}\.html [L,T=text/html]
### BOOST END ###
I've had to comment out RewriteCond %{HTTPS} on [OR]