Yes, you can trust it. The problem comes from Joomla!'s own extensions update code, i.e. what you get when you go to Extensions, Manage Extensions, Update. When Admin Tools loads its control panel we are asking Joomla! "are there any updates?". Joomla! is supposed to connect to our update server (cdn.akeebabackup.com), fetch this information, cache it and tell us if there is an update available. Your host has a firewall which prevents that.
Please ask your host to open communications over TCP port 80 to cdn.akeebabackup.com (note to host: this is a CDN, the hostname resolves to different sets of IPs based on geolocation) and TCP port 443 to www.akeebabackup.com.
Do note that even if you don't do that, you can still trust Admin Tools. It will work just fine. The only think which won't work is update notifications for Admin Tools itself. You will have to check our site about once a month and install updates manually.
Nicholas K. Dionysopoulos
Lead Developer and Director
π¬π·Greek: native π¬π§English: excellent π«π·French: basic β’ π My time zone is Europe / Athens
Please keep in mind my timezone and cultural differences when reading my replies. Thank you!