One specific statement that people are likely referring to when they say things about fibrations and cofibrations being "the same" in spectra is that a homotopy pushout square of spectra is also a homotopy pullback square (considering squares with one corner trivial gives homotopy fibration and cofibration sequences). A brief explanation of this is given by Goodwillie here: Homotopy pullbacks and pushouts of spectra.

If you are interested in a formal statement in one of the model categories of spectra, then you need to look at a proof that these model categories are stable, as defined for instance in Hovey's book Model Categories (Chapter 7). As mentioned here Homotopy limit-colimit diagrams in stable model categories, Hovey explains (Remark 7.1.12) that homotopy pullback squares and homotopy pushout squares coincide in any stable model category. Proofs that the standard model categories of spectra are stable can be found in the basic references for these categories: for instance, for the category of symmetric spectra, Hovey-Shipley-Smith prove this in Theorem 3.1.14 of their paper (Symmetric spectra. J. Amer. Math. Soc. 13 (2000), no. 1, 149–208, available here http://www.ams.org/journals/jams/2000-13-01/S0894-0347-99-00320-3/S0894-0347-99-00320-3.pdf).

One specific statement that people are likely referring to when they say things about fibrations and cofibrations being "the same" in spectra is that a homotopy pushout square of spectra is also a homotopy pullback square (considering squares with one corner trivial gives homotopy fibration and cofibration sequences). A brief explanation of this is given by Goodwillie here: Homotopy pullbacks and pushouts of spectra.

If you are interested in a formal statement in one of the model categories of spectra, then you need to look at a proof that these model categories are stable, as defined for instance in Hovey's book Model Categories (Chapter 7). As mentioned here Homotopy limit-colimit diagrams in stable model categories, Hovey explains (Remark 7.1.12) that homotopy pullback squares and homotopy pushout squares coincide in any stable model category. Proofs that the standard model categories of spectra are stable can be found in the basic references for these categories: for instance, for the category of symmetric spectra, Hovey-Shipley-Smith prove this in Theorem 3.1.14 of their paper (Symmetric spectra. J. Amer. Math. Soc. 13 (2000), no. 1, 149–208, available here http://www.ams.org/journals/jams/2000-13-01/S0894-0347-99-00320-3/S0894-0347-99-00320-3.pdf).