How can one make a connection where flexibility is necessary despite the connection being exposed to physical damage?
As far as I can tell:

Ensure the conduit is not subject to physical damage
Use  liquidtight flexible metal conduit (LFMC) (as opposed to LFNC or LFNMC)


LFNC does seem to be commonly used, probably because it's cheaper, but LFMC is acceptable in this case as well and would stand up better to accidental physical damage.
One of the challenges is even if the electrical line is protected, the refrigerant lines also need to be protected. Commonly they are just two flexible copper lines, one wrapped with foam rubber insulation.

NEC 2011 through 2017:

Article 350
350.12 Uses Not Permitted
LFMC shall not be used as follows:
(1) Where subject to physical damage
350.30 Securing and Supporting
(A) Securely Fastened. LFMC shall be securely fastened in place by and approved means within 300mm (12 in.) of each box, cabinet, conduit body, or other conduit termination and shall be supported and secured at intervals not to exceed 1.4m (4.5 ft).
Exception No. 2: Where flexibility is necessary after installation, lengths from the last point where the raceway is securely fastened shall not exceed the following:
(1) 900 mm (3 ft) for metric designators 16 through 35 (trade sizes 1/2 through 1 1/4)

Article 440 is dedicated to Air-Conditioning and Refrigerating Equipment but doesn't have any mention of physical connection requirements.

As originally stated, the highest chance of damage would come from lawn mowers and trimmers, so the best course of action would be to install the unit in such a way to avoid those things.
This could mean ensuring no grass or weeds grow around the unit by using concrete, gravel or pavers (be sure to use landscape fabric underneath):
 
Or mounting it on the wall, so it's high above the business end of lawn equipment: