Trouble with maintenance staff not answering on-call

Good afternoon
I want to hear some of your ideas on an issue that we currently have at my job. Our maintenance staff is not always answering their phones when they are on-call or even during work hours. They have company issue phones but the problem still exists. Our boss wants to remove the cell phones since they do not answer them at times and do not return calls and sees that as money being wasted by the company. I was tasked to try to find an affordable solution to this problem due to us being on a budget since we are a non-profit organization. Thank you in advance.  

I'd say the bigger problem isn't the waste of money but the not answering the call/serving their coworkers and residents.  If you had a real emergency, would they come out?  
Have you sat down with them and listened to feedback on why they don't answer? It could be a connectivity issue (positive thinking here), in which case you may need to change providers. If it's other issues, like they get so many calls they can't do their jobs, or they are in the middle of something and can't answer, or they get called for every little thing, or... These are common complaints I hear from Maintenance Team Members.  Could there be some compromises and other ways to communicate? 
But if it's a performance issue that you need to make sure your team members understand their job and your policies, well, then there's that. You may need to clearly set parameters, give them a chance to change behaviors, reinforce it. 
On-call is a hard part of the job. Are you showing appreciation for "carrying the phone?" Can you afford a small bonus for the time they are on-call even if they don't get a call?  
I hope this helps!

They do show up to the property in the event of an emergency but sometimes it can be a hassle trying to get ahold of the maintenance on-call. There have been times where the maintenance superintendent or another maintenance from one of our other properties has to go over. I was a maintenance tech and maintenance supervisor myself so I know firsthand how on-call can be. We provide housing to the senior citizens so sometimes they do call for the smallest things that can wait until the morning and it can be a headache at times. Our company currently does not pay extra for being on-call but maintenance start charging the second they answer the phone up until they get back home or to wherever they were when the call was received. They can also leave early or go in late for the same time they spent on the emergency calls throughout the week to avoid them going into OT. Now that I'm in the corporate office and another department (procurement & transportation), I want to change how our maintenance staff gets treated when it comes to being on-call and other areas. My goal is to get them at least a small bonus just for being on-call.  

You are on the right track!  Making sure your team is treated fairly and compensated appropriately - both are a good start.
It may not work for you, but we have economically added answering services to all our property that actually help screen calls to make sure after hours calls qualify as an afterhours emergency, as well as making sure they reach someone on our team.  So if the Maintenance Person on call doesn't answer within a reasonable time, our service goes to that person's supervisor, then to the Regional, and even beyond the Regional.  Is that a possibility for you?  Maybe get rid of moat of the cells, have one that you trade to whoever is on call, and with that savings add an after hours answering service that helps you manage on call? It would reduce unnecessary emergency calls, and save them for the regular workday, which would reduce a little of the stress for your Maintenance team...Just some thoughts!

This is a serious problem, if you have residents with emergencies and on call techs not answering their phones. We have found that many techs don't like having to carry the extra phone around. If you use a service like HelloSpoke, they can use their own phone and their caller ID is blocked when they call the resident back. The service also documents the response time and records the conversation between the resident and the technician. A lot of management companies put incentives in place for their maintenance teams, such as lowest average response time. They also recognize technicians who provide great customer service to the resident. Because the conversation is recorded, it can protect the property and employee from liability (no more he said/she said) and it also makes a great training tool by being able to listen to the calls and coach the employee on their customer service.