Note: A flight plan (FP) is needed for crossing a border, even for VFR.
For US-Canada, the answer is yes to VFR-only planes/flights/pilots. Note: for crossing the ADIZ, that would require DVFR (Defense VFR).
For UK-France, also yes. Note: even a mixed FP is possible (VFR-IFR-VFR). Below are excerpts from UK AIP:
1.11.1 Pilots undertaking Cross-Channel flights are reminded that a flight plan MUST be filed for all flights to or from the United Kingdom which will cross the United Kingdom/France FIR Boundary.
1.11.2 Specific FPL addressing requirements are detailed in paragraph 2 VFR Flight Planning.
2 VFR Flight Plans
22.214.171.124 The CAA have received reports that some VFR flight plans, filed for flights between France and the United Kingdom, have not been received at the UK destination aerodrome. Although these reports are infrequent, they nevertheless identify a significant safety aspect of cross-channel flight planning. The ability of the Air Traffic Service Unit at the destination aerodrome to be aware of an inbound flight is a key factor to alert search and rescue services, when appropriate.
126.96.36.199 Additionally, for flights to/from France, the French Authorities require the frontier crossing point (the UK/France FIR boundary position) to be included in Item 15 (Route) of the FPL. To assist pilots, the UK now includes the ATS route reporting points on the Southern England and Wales 1:500 000 chart. These can be used as a frontier crossing point. A position may also be shown as LAT/LONG, or as a bearing and distance from a route reporting point or navigation aid.
Cap Gris Nez - RINTI
Cap Gris Nez - 51N00130E
Cap Gris Nez - RINTI23005
Cap Gris Nez - DVR16010
As for how that would work, by checking the AIP (1.6 Mechanisms for Filing a FPL), it will be done via 'flightplanningonline' (a fax option is available too).
Also worth submitting, even if not required for a flight to France, is the general aviation report (GAR).
As for the minimum navigation equipment, it is found in GEN 1.5 of each country's AIP. As an example for the UK:
Assuming a single-engine plane flying VFR and not for the purpose of public transport and below FL 245, there doesn't seem to be any special requirement for navigational equipment, except for ILS if flying into "certain aerodromes within Control Zones".
In short, you can fly a bare-bones plane across the Channel if you wish, but since weather can and does quickly change, you need to have an exit strategy if faced with visibility below 5 km and/or lots of unanticipated clouds.