Are splices made with insulated cable connectors required to be inside a junction box?
You'll need a box, and a pretty chunky one at that

Splicing wire this large is indeed done with insulated mechanical splice connectors like the one you linked -- you will need one per wire, and a torque screwdriver or wrench, marked and calibrated in inch-pounds, to install them properly.  As to containing the splices, you will need something larger than a normal junction box -- in fact, what you need is called a pull box, and needs to be sized according to NEC 314.28(A):


  (A) Minimum Size. For raceways containing conductors
  of 4 AWG or larger that are required to be insulated, and for
  cables containing conductors of 4 AWG or larger, the minimum dimensions of pull or junction boxes installed in a
  raceway or cable run shall comply with (A)(1) through
  (A)(3). Where an enclosure dimension is to be calculated
  based on the diameter of entering raceways, the diameter
  shall be the metric designator (trade size) expressed in the
  units of measurement employed.
  
  (1) Straight Pulls. In straight pulls, the length of the box
  or conduit body shall not be less than eight times the metric
  designator (trade size) of the largest raceway.
  
  (2) Angle or U Pulls, or Splices. Where splices or where
  angle or U pulls are made, the distance between each raceway entry inside the box or conduit body and the opposite
  wall of the box or conduit body shall not be less than six
  times the metric designator (trade size) of the largest raceway
  in a row. This distance shall be increased for additional entries
  by the amount of the sum of the diameters of all other raceway
  entries in the same row on the same wall of the box. Each row
  shall be calculated individually, and the single row that provides the maximum distance shall be used.
  
  Exception: Where a raceway or cable entry is in the wall
  of a box or conduit body opposite a removable cover the
  distance from that wall to the cover shall be permitted to
  comply with the distance required for one wire per terminal
  in Table 312.6(A).
  
  The distance between raceway entries enclosing the
  same conductor shall not be less than six times the metric
  designator (trade size) of the larger raceway.
  
  When transposing cable size into raceway size in
  314.28(A)(1) and (A)(2), the minimum metric designator
  (trade size) raceway required for the number and size of
  conductors in the cable shall be used.


Given that you are splicing in this box, and that a 4/3 W/G cable is equivalent to 4 4AWG THHNs (actually a bit less due to the ground being bare, but lets ignore that for now) or 212mm2 of fill, which barely fits down most types of 1" conduit, we wind up needing at least a 6" square box, and likely something more like an 8" square box.  Fortunately, a NEMA 1 pull box of that size (such as this) isn't too hard to get one's paws on -- your local electrical supply house should definitely be able to get one in for you.

Grounding the pull box, by the way, will require you to use either:


a four-way insulated mechanical splice for the ground wires as well as an 8AWG copper pigtail from the insulated splice to the green ground screw in the box.
or a ground bar in the pull box, attached to the box where the green ground screw goes, with the ground wires landed on that.