Hi,

You can remove the deafult=True from the field, or make default=False

customer = fields.Boolean(string='Is a Customer', default=False,
                          help="Check this box if this contact is a customer.")

Then to prevent the saving of record without ticking customer, supplier or your third field. Add a constrain for the model res.partner.

@api.constrains('customer', 'supplier', 'third_field')
def check_ticks(self):
    if not self.customer and not self.supplier and not self.third_filed_name:
        raise ValidationError(_("Customer or Vendor Has to be Ticked..!"))

Full Code:

from odoo.exceptions import ValidationError

class ResPartner(models.Model):
    _inherit = 'res.partner'

    customer = fields.Boolean(string='Is a Customer', default=False,
                              help="Check this box if this contact is a customer.")

    @api.constrains('customer', 'supplier', 'third_field')
    def check_ticks(self):
        if not self.customer and not self.supplier and not self.third_filed_name:
            raise ValidationError(_("Customer or Vendor Has to be Ticked..!"))

Thanks