It is an interesting project, yet, several significant changes need to be made into the design for it to actually work. The most critical part is to keep the impedance of the transmission line (these two wires between the router and antenna connector make up a transmission line) constant. This is in contrast to letting the impedance go uncontrolled (likely very high) by using 2 separate loose wires as shown in the video. at this high frequency such loose wires play a role of inductors and are thus equivalent to a high impedance in series with the load (new antenna). Applying Ohm's law, it is easy to see that the antenna will receive little from the router. Second consequence would be that the signal would be reflected back into the power amplifier, causing additional heating.
There are several aspects to making it all work:
1) ideally, one has to stay with coaxial cables.
2) one can theoretically use twin-lead (i.e. two strictly parallel wires) or twisted wires to do the same but would need to do calculations of correct dimensions to achieve the constant impedance around 50 Ohm:
e.g theory in here:
and an online calculator here:
https://www.easycalculation.com/enginee ... ulator.php
No letting two wires loose - this way your loose wires will be the only "antennas" in the system and very little of the signal will reach the antennas connected to the new connectors...
3) the connectors should ideally be of 50 Om impedance, too. In the video, the connectors are F type and are usually matched to 75 Ohms. Depending on "design", one would easily loose between 0.2 to 1 dB in this.
4) In many routers, there are RF connectors to which one can connect adapters and wires leading to external antennas, rather than un-soldering the coax cables.
5) The antennas would need to be suitable for the frequency ranges used by a router. One band is easier. For higher-end routers, high-gain dual-band antennas would likely be in need (otherwise, it would be questioned why one would need to go for external antennas in the first place).
6) There is a very simple way to gain a few dB for one of the bands (but not for both at the same time!): One can take a metallic sheet and place it behind the router. The critical part is to have the distance between this sheet of metal and the antenna right. It should be about 1/4 of the wavelength (e.g. 0.25*(3e8/2.45e9) = 0.25*12.2cm = 3.1cm for 2.4GHz band - I used 2.45e9 are the middle frequency of the 2.4GHz band).