Strings, once they're tuned, don't have as much of an effect on buzzing vs string height and how the truss rod is adjusted.
It's not too difficult to learn how to set up your own guitar - but it can be a pretty involved process, I would suggest that you start googling and probably even youtubing "setup electric guitar", "how to intonate electric guitar", etc.
Proper string height and neck adjustment should eliminate 90% of your buzz. I've been reading over a Very Thick Book on guitar repair and maintenance, and a direct quote from it is "... Proper setup will eliminate 90% of fret buzzing, but because it is an electric guitar, there will always be some buzzing. This is normal ..."
Because its a Squier strat, there could be some issues with quality in terms of being able to set it up and intonate it well. I'm not saying there will be, but it is possible - Squier is not as well regarded of a brand as, say, Epiphone, in terms of a brand with decent quality yet low price point guitars.
Your strings are broken in - it's time to invest in some tools and get to work on adjusting your guitar. I would suggest a complete set of Allen wrenches (both English and Metric), a good ratcheting screwdriver with an assortment of different bit heads (especially small and different sizes of phillips and flat heads), a clean 100% cotton t-shirt (old or new, just has to be clean), some food grade or better almond oil for fretboard conditioning, and some GHS Fast fret for string lubrication.
Fretboard conditioning doesn't have to be done too often, and there are many different opinions about what works well. I avoid solvents and prefer to use all-natural vegetable-based oils... so no mineral oil, and no lemon oil either, since it usually contains solvents. I've used olive and almond oil with great results - they don't go rancid, so don't believe people who tell you that, either.
GHS Fast fret isn't the only product out there for lubrication, but I prefer it personally. Again, all natural vegetable-based oil. I put it on my strings before I play, and a little bit on my fingertips as well. That way, if I forget to wipe my strings down after I play, it doesn't impact them as much, since the oil helps form a barrier between the strings and whatever dirt and oils may be on my hands (use the cotton t-shirt to wipe your strings down after every use and double the lifespan of your strings!).
I also prefer Elixir strings, mostly because they're coated and last a long time before they go dead. I hate changing strings unless I have to, and between using Elixirs and the Fast fret and wiping down, etc, even with heavy playing I can go months without having to change strings!
Free tip - if you rub the tip of a pencil (normal, not mechanical) along the grooves of your nut (where the strings go) it helps them slide smoothly, which means you tune more accurately and without that pinging sound you get sometimes when the string catches and releases while tuning.
Good luck!
Saul