Home setup
You can host ZNC (free IRC bouncer/proxy software) on a always on computer (for example a raspberry pi), use free Letsencrypt SSL certificates and setup free DDNS (for example duckdns.org) and configure a port forward on your router so you can connect to your bouncer from mIRC from wherever you are in the world:

Code
/server -m myname.duckdns.org +1337 znc/freenode:mypassword


Where:
- ‘myname’ is the name you register with duckdns.org
- ‘+1337’ denotes the port forward you opened on the public interface, use a + for ssl
- ‘znc’ denotes the username you configured in znc software
- ‘/freenode’ denotes the network you wish to connect to (but you have to set all that up in znc first)
- ‘:password’ is the password you configured for the ‘znc’ username

You can use palaver on IOS to connect to myname.duckdns.org like you connect with mIRC from your computer, to check up on your messages.
I’ve no experience with android apps in the matter, but someone else might know.

Free ZNC provider
Tools from LunarBNC has made https://lunarbnc.net, afaik is free and better than decent. (What you are looking for). You may wish to try it out!

Paid BNC provider
I’m now using irccloud.com. I pay a few euro’s/month for a very stable connection. Also, it comes with android and iPhone apps, so you can check your messages when away from your computer. You can use a webbrowser as an interface to your IRC network connections (and what a nice interface it is!) or configure it so that you can use any other irc client. (It will act as a bouncer/proxy)

Paid VPS (Virtual Private Server)
I have experience with Strato and have rented a few servers in the past for as little as €9,-/month. Connect to it with putty on windows or your favourite ssh client and configure the VPS as explained in the home setup. Obviously you can skip the duckdns.org part! (For as little as 4€/year you register your domain name)

GNU Terry Pratchett - Looking for a mIRC help channel -> Check #mircscripting @ irc.swiftirc.net