Trouble accessing only one secure website turned out to be a firmware issue with LinkSys router. Upgraded to 1.50.18 and all is well.
http://www.texasworkforce.org/ui/tax/emtaxinfo.html
https://m06hostp.twc.state.tx.us/TAXWEB/TaxMain
After receiving the following error while trying to upgrade the firmware using the web interface, I got my hands on
befsx41_Tftp.exe and upgraded without any trouble:
Warning Upgrade Pattern Error
( BEFSX41 )