Squegging results when the time constant and impedance of the bias network, and the feedback magnitude, are such that rectification occurs, pushing the amplifier into such deep class C that loop gain, averaged over the cycle, drops below unity, and it rings down. The network then recovers, slowly, and the process restarts.
It's a blocking oscillator, "with extra steps". That is, the overall behavior is that of a relaxation oscillator, where it just so happens the regenerative (negative resistance) phase occurs with high-frequency oscillation, but the overall effect is the same.
That there are two different time constants (base and emitter) is peculiar. This isn't necessarily a problem: they act more or less in parallel, so it's a pole-zero in the response (assuming a linearized operating point, which we're only occasionally crossing in actual operation anyway), not a double-pole (that would cost phase margin and suggest oscillation). They should be equal (i.e., C1 * (R2 || R3) = R1 * C6, but probably C6 has to be larger because r_e is a variable term in its response as well).
Feedback amplitude must be reduced to just what is required; with no loss in the circuit (no load -- remember, SPICE components are ideal down to numerical error*), power only has one place to go, the base (well, and whatever the scope probe is taking).
*But do check the properties of everything: LTSpice has a bad habit of hiding default-value parasitic parameters behind dialogs.
Why SPICE hasn't captured the squegging, dunno, but there may be further differences between your actual circuit. Note that supply impedance matters, too (bypass it just in case). Ground-return and series component inductances may matter (at least, probably not a whole lot for frequencies a 2N3904 can reach).
Huh. There are also two degenerate nodes in your simulation. RSHUNT or LT's modified solver (see settings) are probably enough to deal with that, but in general it is a bad idea to connect capacitors in series in SPICE. (These are C1, C4/C5, and C2/C3, C5 via L2.) The usual error is "singular matrix" (the matrix version of divide-by-zero), because the DC voltage on those nodes has no solution, it can be literally anything; tweaks, as above, can determine it, but leaving the matrix near-singular often costs simulation speed.
Tim