I was having similar problems from my laptop, then discovered why I wasn't seeing the link. I'm posting this in the hope that it helps others in a similar situation.
Within my browser (Firefox), I had Javascript disabled via an extension called NoScript, an adblocker (uBlock Origin) turned on, and pop-up windows disabled (via preferences in Firefox). When I allowed the Boost site to work with all three of these, then the clickable button for livechat appeared on the main "help" page on Boost's site, and the livechat session began in a popup window. Other browsers for most systems (Chrome, Safari, etc.) often have similar settings to the above, which can be tweaked accordingly.
(Side note 1: It seems that Boost / Telstra have upgraded their livechat platform since last time I used it - I don't remember being bamboozled in getting the old livechat system to work, as I temporarily was with this new one
Side note 2: It seems like a design flaw in Boost's help page as at July 2017 that there is not a workable fallback in place to detect that Javascript / pop-ups etc are being blocked in the user's browser, and prompting them to enable these for the site to work fully - if the user is not aware of this, they have no obvious way of knowing that their browser settings need to be adjusted).
Cheers,
KT.
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