To start, let's talk about what this equipment is.
The Verizon ONT, pictured at https://imgur.com/gDu4Ltr, is the termination point of Verizon's optical network. This breaks out the fiber network into services that are more useful to the people at the service location; the internet, the phone line(s), and the cable TV service.
The telephone network interface pictured at https://imgur.com/DffmBzh is the termination point for the old Plain Old Telephone System (POTS) network that provided phone service to the location
The device pictured at https://imgur.com/kiiKv6U is a telephone breakout box, most likely used to direct incoming signals to multiple phones so that the phone could be answered from multiple rooms.
So, all that being said, the very first question you should ask is, "Do I own any of this equipment?" The answer is most likely not. Verizon, formerly Bell Atlantic, most likely owns all that equipment, down to the cables run through the walls. You should call them to verify who owns the equipment before proceeding.
If you do own at least the cables, that still may not mean it is viable to convert your existing RJ-11 jacks to RJ-45. Phone lines, and RJ-11 connectors, uses four lines, while Cat5e uses eight. So most likely the Cat5e cable is being terminated at two physically separate locations. If the entire cable is run through the junction box behind an RJ-11 plate it may be possible to re-terminate it as RJ-45 there, but the other jack will be dead.
You should verify how many twisted pairs are being run to each junction box after verifying you own the cables, if there are four twisted pairs it is possible to repurpose it for ethernet use.