If Sweden wants to export JAS39 to a 3rd country, will they have to get permission from the US govt.?
The most correct answer I can give is, It depends. More specifically, it depends on the specific components that are US sourced and it depends on who the ultimate end user is. But the most likely answer is yes.
All technology exports from the US fall under one of two sets of rules. They are the International Traffic in Arms Regulations (ITAR) controlled by the Dept of State and the Export Administration Regulations (EAR) controlled by the Dept of Commerce.
The US engines (F404/F414) that are the source of parts for the Volvo engines are military products controlled by the ITAR. When parts are exported separately from the complete engine, the export controls are set at the part level. So some parts may be controlled differently than others. But, in general any part controlled by the ITAR will require a license from the Dept of State before it can be exported. It is also possible that some components may be EAR controlled and require a license from the Dept of Commerce.
The licenses for export do contain restrictions on use, the end user, and reexport. That is typically spelled out in the individual license. Also, the 'technology' needed to transform the parts into an engine is controlled separately. That includes drawings, software, tooling, etc.
The question of reexport is covered in the ITAR under 22 CFR §123.9 Country of ultimate destination and approval of reexports or retransfers, and 22 CFR §123.10 Non-transfer and use assurances.
Specifically, from §123.9

(a) The country designated as the country of ultimate destination on
an application for an export license, or in an Electronic Export
Information filing where an exemption is claimed under this
subchapter, must be the country of ultimate end-use. The written
approval of the Directorate of Defense Trade Controls must be obtained
before reselling, transferring, reexporting, retransferring,
transshipping, or disposing of a defense article to any end-user,
end-use, or destination other than as stated on the export license, or
in the Electronic Export Information filing in cases where an
exemption is claimed under this subchapter, except in accordance with
the provisions of an exemption under this subchapter that explicitly
authorizes the resell, transfer, reexport, retransfer, transshipment,
or disposition of a defense article without such approval. Exporters
must determine the specific end-user, end-use, and destination prior
to submitting an application to the Directorate of Defense Trade
Controls or claiming an exemption under this subchapter.
(b) The exporter, U.S. or foreign, must inform the end-user and all
consignees that the defense articles being exported are subject to
U.S. export laws and regulations as follows:
(1) The exporter must incorporate the following information as an
integral part of the commercial invoice, whenever defense articles are
to be shipped (exported in tangible form), retransferred (in tangible
form), or reexported (in tangible form) pursuant to a license or other
approval under this subchapter:
(i) The country of ultimate destination;
(ii) The end-user;
(iii) The license or other approval number or exemption citation; and
(iv) The following statement: “These items are controlled by the U.S.
government and authorized for export only to the country of ultimate
destination for use by the ultimate consignee or end-user(s) herein
identified. They may not be resold, transferred, or otherwise disposed
of, to any other country or to any person other than the authorized
ultimate consignee or end-user(s), either in their original form or
after being incorporated into other items, without first obtaining
approval from the U.S. government or as otherwise authorized by U.S.
law and regulations.”