You have two options. First if the labor law requires they be paid, then you must pay them for the past instances. Let them know how paying this impacts things you will be no longer be able to pay for such as conferences, new equipment, etc. Be very transparent on where the money to pay this came from.
Going forward, you can budget the money to pay this as part of their normal salary. Let them know there will be no salary increases next year (or whatever budget items you have to cut to make this work for your company) as you have to absorb this major change to your budget. Show them the budget calculation. Assume a low figure for the salary calc (and make sure they know you are low balling) but say 100 people who get paid an extra hour 5 days a week every week is a lot of money even if you lowball the per hour salary.
Or you can tell them specifically and in writing that the policy is that breaks will be taken unless approved in writing in advance by a manager. Then the first time after that someone works through a break without written approval, you write them up for insubordination and let them know that three strikes and they are out (i.e. fired). If you have an HR team, have them draft the policy as they should be aware of the legal requirements. If not have labor lawyer draft the policy. Do not write this policy without consultation with someone who is up on labor laws.
It is tough to have to be harsh, but this is a huge budgetary issue for you and it is something that warrants firing when they disregard your specific workplace rules on breaks. But don't make the threat unless you take action when someone defies the policy just as you would if they deliberately smashed some equipment or stole from the company. This is that level of serious. (It is stealing from the company to get unauthorized payments.) You will lose some people because they don't want to be adults. Fine, good riddance.
Further not taking breaks is counterproductive just like working overtime is. They actually get less work done over time because they are tired. Remind them of that.