The traditional practice in Chabad was to only start putting on Rabbeinu Tam Teffilin with permission from the Rebbe* (basically establishing that they had achieved this requisite level of piety), and typically after marriage. However, the Alter Rebbe of Lubavitch writes in the siddur that "the custom is to put on Rabbeinu Tam Teffilin" basically implying that everyone should do it.
The most recent Lubavitcher Rebbe established in 1976 that this should begin at Bar Mitzvah, and encouraged everyone to do it. You can see the point summarized here:
But, comes the protest, in the previous generation there were many truly G-d-fearing Rabbis and Yeshivah heads who did not put on two pair of tefillin. Why should we be different? We must recognize, however, that their circumstances were very different from our own. Life was a continuous struggle, to eke out the most meager of existences a task of the greatest difficulty. A pair of good tefillin was very expensive, and it was literally impossible to find enough money to buy a second pair of tefillin (Rabbeinu Tam). Hence, their inability to buy and put on Rabbeinu Tam tefillin did not indicate any deficiency in their fear of G-d. Rather, they labored mightily just to find a valid reason to justify the fact that they could not put on Rabbeinu Tam tefillin. But had they had the means, they most certainly would have.
Today, however, we have no such excuse. Think of all the luxuries we take for granted, all the indulgences we so frivolously waste money on. May we suggest foregoing the latest-model car and using the money saved to buy Rabbeinu Tam tefillin? No one need deprive himself of any necessities in order to buy a pair of Rabbeinu Tam tefillin. It is but a small price to pay to fulfill the dictates of the Shulchan Aruch and be a "G-d-fearing" person.
And let no one piously object with the claim that he does not wish to differ from the customs of his forebears who were unable to put on Rabbeinu Tam tefillin. Do such people so zealously follow all the practices and customs of their forefathers? Their ancestors did not waste precious time and resources on such things as reading newspapers and other trivialities. Why is it only when it comes to excusing oneself from doing good things, such as putting on Rabbeinu Tam tefillin, that such zealots suddenly awaken with their pious claims of following in their forebears’ footsteps? Do not use our holy ancestors as an excuse for personal shortcomings!
Regarding the issue of מחזי כיוהרא (which seems to be the issue in 34:3) - see what the Pischei Teshuva says about that - it depends on the prevalence of the practice.
* I encourage anyone interested in the history to watch that video, it is very clearly laid out.