Any custom code you write (or you) you do to display this output "securely" within the browser with SSL/TLS will be open to MITM attacks (see JavaScript cryptography considered harmful): browsers don't give you any way to verify the authenticity of the code you're running (going through all the JavaScript code whenever you use it is quite unrealistic).
You get a few options:
If you don't expect the output to be read from a browser, you can make your server encrypt the data, then you can save it from your browser and decipher that file using another application.
Some shared hosting services will still serve your content at https://www.yourserver.example/ using their own certificate (only valid for www.yourhost.example, but not www.yourserver.example).
If this works in your case, and if you can trust that the certificate presented to you by the server is realistically that of the server on which your service is hosted (e.g. a cert that would be valid for www.yourhost.example), you can add a permanent security exception in Firefox (for example). This is usually not acceptable if your site is for more public use, but if you can judge that you can trust that server certificate, you will still have all the benefits of SSL/TLS when connecting to that server. This is clearly not ideal and require some of your own judgement when setting such exception. In your case, if the certificate you get for https://www.yourserver.example/is valid for secure1.hostgator.com (or something like that), it would seem reasonable to assume you're connecting to your shared server.
Use your host's shared SSL facility (as they seem to provide one). The URL won't look like it's yours, but it should at least be secure without any specific exception, and it's fine for personal access.
EDIT: To clarify the point about custom code.
Most modern browsers have a key/cert container (e.g. Software Security device on Firefox at least) where private keys can be used for SSL/TLS client authentication.
In this case, the private key is always controlled by the browser itself, and never given away to any page or script. However, browsers generally don't expose the private key in the same way for use by JavaScript so as to be able to decrypt some content without giving full access to the private key itself: in this case, you could never be certain that the key isn't sent to some other server, especially in a context where a MITM attacker would have been in a position to alter the script (which is unverifiable, since you can't really see the code you're running on a page).
There is a Mozilla JavaScript extension (signText) that should protect the key for signing the correct way, but there's nothing to decrypt. There are also on-going efforts to make more cryptographic operations available, but this isn't ready at this stage (it's not clear whether it will be implemented in IE either).
secure access to the text output by three PHP scripts--> SSL and a VPS (virtual private server) which usually means a personal IP address. I can help you get all of that for approximately 3 dollars monthly, plus a one-time SSL certificate (purchase one every few years). You don't need an SSL certificate to use encryption, though! You can create a self-signed one for free. – ionFish Jun 23 '12 at 19:19