viander, having a custom domain is not only recommended practice, it also brings other benefits, such as being able to connect an SSL certificate. Did you know that if you subscribe to the Optimal plan for a year, you get a free domain (along with other cool stuff - for example, the removal of ads on the website)? See more details here: https://www.ucoz.com/pricing
Once you have a domain you can try using this service to get an SSL certificate for free: https://zerossl.com/ although I haven't tested this on uCoz, so it might not work (the free SSL doesn't work with all web servers).
hey i'm joe and i do not work for the company anymore, sorry for the late replies icon by ch-chau