How can I adapt old thick wiring entering my 30 yr old furnace to fit into the clip on slot on modern day circuit breaker switch?
This is what unit mount breakers are for

Most modern circuit breakers are designed to attach to a panelboard's busbars, hence the clip.  However, you don't want that since these breakers are providing subdivided protection per NEC 422.11(F)(1), with fat branch circuit wiring coming in and no busbars in sight within the enclosure:


  (F) Electric Heating Appliances Employing Resistance-Type Heating Elements Rated More Than 48 Amperes.
  
  (1) Electric Heating Appliances. Electric heating appliances employing resistance-type heating elements rated
  more than 48 amperes, other than household appliances
  with surface heating elements covered by 422.11 (B), and
  commercial-type heating appliances covered by 422.11 (D),
  shall have the heating elements subdivided. Each subdivided load shall not exceed 48 amperes and shall be protected at not more than 60 amperes.
  
  These supplementary overcurrent protective devices
  shall be (1) factory-installed within or on the heater enclosure or provided as a separate assembly by the heater
  manufacturer; (2) accessible; and (3) suitable for branch-
  circuit protection.
  
  The main conductors supplying these overcurrent protective devices shall be considered branch-circuit conductors.


What you need instead is a unit mount breaker, sometimes called a cable-in/cable-out breaker -- these circuit breakers are designed to mount directly into a box, usually through a mounting scheme such as the ubiquitous DIN rail or a set of panel mounting clips, instead of to a panelboard busbar, and have wire lug terminals on both sides of the breaker.  An example is depicted below.



However, you need a breaker listed to UL 489 for this application.  Look for something from the Square D QOU or Eaton QC lines, for instance, although several other UL 489 listed DIN rail breaker lines are available as well -- you'll need to order them in from an electrical supply house though, along with the appropriate mounting hardware if you are changing breaker types (a like-for-like replacement should fit the existing mounts though).