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.