That's very strange. Even if your web host is incompatible with Safari (which in itself is rather odd), I'd think that they might want to make some changes to ensure that it is compatible, given the number of Safari users out there. (Safari is currently second only to Chrome in the number of users, according to StatCounter, August 2019.)
In any case, looking at the results of an ICANN lookup for 'juicebox.net', I see that the nameservers are ServInt which looks to be an enterprise solution rather than a 'regular' web host. (As you have probably gathered, I personally have no direct dealings with our web host.)
Having said that, I've never heard of a web host that claims to be incompatible with Safari so yours appears to be the exception to the rule and I expect that the majority of web hosts will support both Safari and Juicebox-Pro's Download Button (as long as PHP 5.2.0 or later is installed).
I wish you well in your search for a compatible web host but I really don't think that you should have to look too far.
I'm very surprised by your web host's admission and refusal to try to fix the problem (which could just be a tweak to their Mod_Security settings which they seem to be unwilling to change).
I would hesitate to recommend a web host, though, as one person's experiences may differ greatly to another's.
However, I have free accounts with a few web hosts and I have uploaded a test Juicebox-Pro gallery (featuring the Download Button) to each of them so that you can check out the Download Button's functionality in your Safari browser.
AwardSpace: [Link removed.]
ByetHost: [Link removed.]
x10hosting: [Link removed.]
Please be aware that all the galleries above are on free accounts and you usually get what you pay for.
Paid accounts usually offer more features, greater stability and speed and better support.
Also, please note that I'm not actually recommending any of these web hosts.
This is really just a test to (hopefully) demonstrate that most web hosts will support Safari and Juicebox-Pro's Download Button.
Please let me know how you get on.